For some reason in TSL I'll sometimes get conversations (or locked into this somehow?) where all of the NPC dialogue is cycled through in about half a second, so it's a flicker and then you have to respond with whatever. Sometimes it's hard to even figure out what you're supposed to be responding TO 8(

I've sometimes been able to reset by reloading an old save file or shutting off the game and restarting, but is this a common problem? Is there a quick fix somewhere? Does anyone know?

This is a common problem, although it is often mod induced. However, many conversations in TSL are buggy in this way out of the box, especially T3-M4's dlg.

As you have figured out, saving often and in different slots can be helpful.
Make sure your game is updated, that you installed it correctly for your OS, and then maybe choose some mods that fix that kind of stuff. TSLRCM would be my recommendation.